HONG KONG—Thousands of people converged on the headquarters of Hong Kong's government on Saturday after televised clashes between students and police energized protesters seeking democracy in the city.

Earlier in the day, police arrested dozens of student protesters who had scaled fences at government headquarters overnight following a week of pro-democracy protests. Police had used pepper spray on the students, which sparked thousands of city residents to come to the site and watch the arrests from overhead skywalks...